A power saving for sets avoiding polynomial differences

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2026-10-11

Math Digest

Combinatorics

Family 182 · Combinatorics
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-11

1. The question

How many integers can be chosen from $1,\ldots,N$ if no two chosen numbers differ by a nonzero square? More generally, what if differences are forbidden whenever they equal a value of a fixed polynomial?

Classical recurrence theorems say that the density of such a set must tend to zero, provided the polynomial has no local congruence obstruction. The quantitative challenge is how quickly it must vanish. A fixed power saving would bound the size by $CN^{1-c}$ for some constant $c>0$ , crossing a longstanding barrier between subpower decay and polynomial decay.

The three manuscripts claim this improvement for squares, for every intersective polynomial of fixed degree, and for suitable polynomials evaluated only at primes. The prime-argument result uses a separate new zero-free theorem for Dirichlet $L$ -functions. Its dependency and its stronger local hypothesis are essential parts of the statement.

2. Background and essential definitions

Write $[N]=\{1,\ldots,N\}$ and

A-A=\{a-b:a,b\in A\}.

A set is square-difference-free when $A-A$ contains no positive square. Since differences occur with both signs, this is equivalent to forbidding a nonzero square as an absolute difference.

An integer polynomial $h$ is intersective if it has a root modulo every positive integer $q$ : for each $q$ , some integer $r$ satisfies $h(r)\equiv0\pmod q$ . The roots may vary with the modulus, and $h$ need not have a rational root.

This condition is necessary for a general density theorem. If $h$ has no root modulo some $q$ , all multiples of $q$ form a positive-density set whose differences avoid every value of $h$ . Squares are intersective because zero is a root modulo every modulus.

When arguments are restricted to primes, the condition becomes prime-intersectivity: for every $q$ , there is a root $r$ with

\gcd(r,q)=1, \qquad h(r)\equiv0\pmod q.

Such a root is called a unit root. It belongs to a residue class compatible with primes not dividing the modulus. The polynomial $(x-1)^2$ has this property, using $r=1$ . The polynomial $x^2$ does not: modulo a prime, its only root is zero.

A power-saving estimate has the form $|A|\leq CN^{1-c}$ , so the density is at most $CN^{-c}$ . This is stronger than

N\exp\bigl(-a\sqrt{\log N}\bigr),

because a fixed power saving corresponds to $\exp(-c\log N)$ . The constants are fixed before $N$ grows.

3. Development history and previous partial results

Furstenberg and Sárközy independently proved that every positive-density set of integers contains two elements with nonzero square difference. Furstenberg used ergodic theory; Sárközy used Fourier analysis. Kamae and Mendès France extended the density conclusion to nonconstant intersective polynomials, matching the necessary congruence condition.

Quantitative bounds then improved through increasingly refined density-increment arguments. Pintz, Steiger and Szemerédi introduced a stronger iteration for square differences. Bloom and Maynard improved the logarithmic decay using additive-energy estimates for rational frequencies.

Green and Sawhney obtained

|A|\ll N\exp\bigl(-a\sqrt{\log N}\bigr)

for square-difference-free sets. Their arithmetic level inequality connected rational Fourier coefficients with product-space methods. For general intersective polynomials, Lucier developed auxiliary polynomials adapted to arithmetic progressions, and Rice and Arala strengthened the quantitative bounds. Adajar and collaborators extended the newer method to obtain density decay $\exp(-a(\log N)^\mu)$ for every fixed $\mu<1/2$ .

Lower-bound constructions show that square-difference-free sets can still be large. Ruzsa and later authors built examples with power-law size; the manuscript cites Krachun’s lower exponent exceeding $0.7527$ . Thus even a power-saving upper bound leaves a substantial exponent gap.

The prime-argument problem has its own history. Sárközy treated differences $p-1$ and $p+1$ , and Green later obtained a power saving for these linear examples. For nonlinear prime-intersective polynomials, Rice proved quantitative logarithmic bounds and recorded the qualitative characterization through unit roots. M2 claims a fixed power saving in this nonlinear setting, using the square and polynomial framework together with its stated analytic companion input.

4. What the new manuscripts claim

M3: square differences

M3 states that absolute constants $c>0$ and $C<\infty$ exist such that every square-difference-free $A\subseteq[N]$ satisfies

|A|\leq CN^{1-c}.

The exponent is described as extremely small, with no useful numerical value or optimality claim. The result also gives a polynomial bound for square-difference colorings: if

N>(Cr)^{1/c},

every $r$ -coloring of $[N]$ has same-colored elements $a,a+d^2$ with $d\geq1$ . Only the two endpoints must share a color; $d$ has no color requirement.

M1: all intersective polynomials of a fixed degree

For every degree $k\geq2$ , M1 claims an exponent $c_k>0$ common to all intersective integer polynomials of degree $k$ with positive leading coefficient. For each fixed such $h$ , there are constants $C_h,N_h$ such that

(A-A)\cap h(\{1,2,\ldots\})\subseteq\{0\} \quad\Longrightarrow\quad |A|\leq C_hN^{1-c_k}

for $N\geq N_h$ . The leading constant and starting threshold may depend on every coefficient of $h$ . Only the saving exponent is uniform across a fixed degree.

Zero values are explicitly permitted in the intersection: a polynomial root contributes no forbidden nonzero difference. Arguments are positive integers, and differences are ordinary integer differences, not merely residues modulo a chosen modulus.

M2: polynomial values at primes

For each fixed prime-intersective integer polynomial $h$ of degree at least two with positive leading coefficient, M2 claims constants $c_h>0$ and $C_h\geq1$ such that

(A-A)\cap\{h(p):p\text{ prime}\}\subseteq\{0\} \quad\Longrightarrow\quad |A|\leq C_hN^{1-c_h}

for every positive integer $N$ . Here the exponent may depend on the whole polynomial, not only its degree.

M2 uses the companion claim that every Dirichlet $L$ -function is zero-free in $\operatorname{Re}s>7/8$ , with the principal pole at one allowed. Dirichlet $L$ -functions encode arithmetic progressions through multiplicative periodic coefficients; control of their zeros yields estimates for primes in residue classes. This is the specific new input explained in family 003, rather than a classical zero-free estimate silently assumed by the prime argument.

5. Why the advance matters

Importance: 5/5. The square-difference problem is a foundational quantitative recurrence problem. Crossing to a fixed power saving changes the scale of the answer, and extending that conclusion to the full local-obstruction class of integer polynomials greatly broadens its reach.

The fixed-degree uniformity in M1 is another substantive feature. Polynomial coefficients can create very different congruence behavior. The claim allows those differences to affect constants and starting scales while keeping one positive exponent for each degree.

The prime result is stronger in a different direction: forbidding only values at prime arguments excludes fewer differences, so avoiding sets could potentially be larger. The manuscript claims power decay even under that weaker avoidance condition, for the appropriate unit-root class and using its stated analytic dependency.

These bounds remain far from an optimal exponent. Their importance lies in establishing a new qualitative rate of decay, not in predicting the largest avoiding set accurately for a practical numerical range.

6. The main idea

Measure more than density

Classical density-increment arguments locate an arithmetic progression where the set becomes denser, then repeat. The manuscripts instead construct a nonnegative multilinear quantity $Y(N,A)$ that dominates a fixed power of density:

\left(\frac{|A|}{N}\right)^d\leq Y(N,A),

where $d$ is a fixed even number of slots in an auxiliary construction. They seek a recurrence forcing $Y$ to decay as a power of $N$ .

The construction works first over finite fields. It chooses probability laws on tuples of residue labels so that selected differences are polynomial values, except for a small controlled exceptional part. The laws also satisfy reflection positivity: splitting the tuple along prescribed reflections gives positive semidefinite coupling matrices.

Repeated Cauchy–Schwarz inequalities across these reflections yield a norm-like estimate. This remains valid for signed inputs, which is necessary because the Fourier approximations used later need not be nonnegative.

Connect local residues with actual integer values

A Fourier lift retains selected rational-frequency coefficients of the indicator of $A$ . It places the integer set into a product of residue spaces while preserving its mean density.

Residue constraints alone do not ensure an actual square or polynomial difference in the integers. The papers therefore construct a signed kernel supported on genuine polynomial values. Its Fourier transform approximates the pair distribution prescribed by the tuple law.

If $A$ avoids those values, the kernel’s contribution on appropriate ordered pairs from $A$ vanishes, irrespective of the signs of its weights. This cancellation controls a positive portion of the multilinear expression. Other terms are compared with the same kind of expression on shorter intervals, and exceptional contributions are kept summable.

Preserve the problem when changing scale

For squares, restricting to a progression of square step preserves square-difference avoidance after rescaling. General polynomials change under progression restriction, so M1 organizes them into a compatible family of auxiliary polynomials.

A normalization at chosen local roots makes the structural parameters determining the exponent depend only on the degree. Weighted tuple laws transform compatibly with the auxiliary family, while the signed kernels reproduce their changing polynomial-value distributions.

M2 must additionally restrict arguments to unit residue classes and realize its kernel on values at actual primes. The companion zero-free theorem supplies the needed prime-distribution estimates on the major arcs, the neighborhoods of simple rational frequencies. Other frequencies are controlled by exponential-sum methods. The final contraction follows the same broad architecture, with constants depending on the fixed polynomial.

7. Scope, limitations, and open questions

The power-saving exponents are positive but not optimized or numerically useful. The square problem still has a large gap between known construction exponents and the claimed upper exponent near one.

M1 fixes the polynomial before $N$ grows. Uniformity of $c_k$ does not make the constants uniform over arbitrarily large coefficients. M2 has a different guarantee: even its exponent may depend on $h$ .

Prime-intersectivity is stronger than ordinary intersectivity. In particular, M2 does not apply directly to $h(x)=x^2$ and therefore does not claim the same theorem for forbidding only prime squares. It does apply to $(x-1)^2$ , giving a power saving when the forbidden differences are $(p-1)^2$ .

The prime theorem depends on the specified zero-free companion. M1 and M3 do not require that prime-distribution input. Keeping the dependency separate is necessary to understand what each manuscript establishes within the collection.

Finally, the conclusions force one forbidden difference, not a long polynomial progression or a monochromatic polynomial configuration with every parameter sharing the same color. Such extensions would need additional statements.

8. Sources and further reading

M3 develops the square-difference framework. M1, dated October 5, extends it to general intersective polynomials with a degree-dependent exponent. M2, also dated October 5, adapts the method to prime arguments.

Primary predecessors include Bloom–Maynard, Green–Sawhney, Adajar and collaborators, and Rice on prime-intersective polynomials. The prime-distribution dependency is the September 30 zero-free manuscript, discussed in family 003. M1 and M3 also explain the earlier recurrence theorems and the reflection-positivity background.

Source manuscripts

M1. OpenAI, A power saving for intersective polynomial differences with an exponent depending only on the degree (October 5, 2026).
M2. OpenAI, A Power Saving for Polynomial Differences at Prime Arguments (October 5, 2026).
M3. OpenAI, A power saving for square-difference-free sets (September 24, 2026).

Sources consulted on 11 October 2026 at repository revision fd4aeeb. Download the manuscript citations.